Stability properties and gap theorem for complete f-minimal hypersurfaces

2013/07/18 by Xu Cheng, Cheng, Xu, Detang Zhou +1 · 1 citation
Mathematics · #58J50 #Bar (unit) #Combinatorics #Computer science #Cylinder #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Holomorphic and Operator Theory #Hypersurface #Mathematical analysis #Mathematical physics #Mathematics #Physics #RADIUS #Stability theorem #math.DG #msc:58J50

paper · pdf · doi:10.48550/arxiv.1307.5099

published in arXiv (Cornell University) (Cornell University) · 23 pages

arxiv created 2013/07/18 · openalex publication_date 2013/07/18 · arxiv updated 2013/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

In this paper, we study complete oriented f-minimal hypersurfaces properly immersed in a cylinder shrinking soliton (\mathbbSⁿ× ℝ, g, f). We prove that such hypersurface with L_f-index one must be either \mathbbSⁿ×\0\ or \mathbbS^n-1×ℝ, where \mathbbS^n-1 denotes the sphere in \mathbbSⁿ of the same radius. Also we prove a pinching theorem for them.

Citations

• Hamilton’s Ricci Flow
• Eigenvalues of the drifted Laplacian on complete metric measure spaces
• Local Rigidity Theorems for Minimal Hypersurfaces
• f-Minimal Surface and Manifold with Positive m-Bakry–Émery Ricci Curvature

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